Spreading and shortest paths in systems with sparse long-range connections.

Area of Science:

• Complex systems
• Network science
• Statistical physics

Background:

• Spreading processes, such as disease or fire propagation, are fundamental in various scientific fields.
• Understanding shortest-path distances is crucial for analyzing network connectivity and efficiency.
• Small-world networks, characterized by a low density of long-range connections, exhibit unique topological properties.

Purpose of the Study:

• To investigate spreading dynamics and shortest-path distances in d-dimensional systems with long-range connections.
• To derive exact formulas for the volume covered by a spreading process over time.
• To analyze the relationship between Euclidean distance and average shortest-path distance in these networks.

Main Methods:

• Exact calculation of the volume V(t) covered by the spreading quantity in infinite systems.
• Derivation of the average shortest-path distance l(r) as a function of Euclidean distance r.
• Analysis of the behavior of shortest-path lengths relative to a characteristic length r(c).

Main Results:

• The volume V(t) covered by spreading was exactly calculated for all dimensions.
• A critical length scale, r(c), was identified, determining the crossover in shortest-path distance behavior.
• Shortest-path distances scale linearly with Euclidean distance for r < r(c) and plateau for r > r(c).

Conclusions:

• The characteristic length r(c) diverges logarithmically with system size L for any non-zero connection probability p.
• This finding has implications for understanding information or disease spread in large, sparsely connected networks.
• The study provides a theoretical framework for analyzing transport and connectivity in complex systems.